On the classification of real forms of non-Abelian Toda theories and W-algebras

We consider conformal non-Abelian Toda theories obtained by Hamiltonian red uction from Wess-Zumino-Witten models based on general real Lie groups, We study in detail the possible choices of reality conditions which can be imp osed on the WZW or Toda fields and prove correspondences with sl(2,R) embed dings into real Lie algebras and with the possible real forms of the associ ated W-algebras. We devise a method for finding all real embeddings which c an be obtained from a given embedding of sl(2, C) into a complex Lie algebr a. We then apply this to give a complete classification of real embeddings which are principal in some simple regular subalgebra of a classical Lie al gebra. (C) 1999 Elsevier Science B.V.